Blurry and noisy imagery is a ubiquitous problem for cameras and sensors across the spectrum of applications, including consumer photography, surveillance, computer vision, remote sensing, medical, and astronomical imaging. Simple methods to obtain sharp imagery are a valuable asset to all of these applications.
Although there are many sources of image blurring, blurring can be modeled by a single blur kernel, which is also called a point spread function (“PSP”). An equation for describing the observed blurry and noisy image as a function of the underlying sharp image isb=hi+n,  (1)where b is the observed blurry image, h is the blur kernel,  is a convolution, i is the sharp image, and n is noise.
As shown, by way of example, in FIGS. 1A-1C, step edges in the real-world are convolved with a blur kernel to produce the blurry edge profiles seen in imagery. FIGS. 1A, 1B, and 1C are pictorial representations of an illustrative step edge, an illustrative blur kernel h for Eq. (1), and an illustrative edge profile, respectively. It is reasonable to assume the edges in the real, continuous scene being imaged are step edges of discontinuous intensity, such as shown in FIG. 1A. However, what is observed is a blurry image containing edge intensity profiles such as shown in FIG. IC.
In the overwhelming majority of real-world situations, only the blurry image is available. Solving for the sharp image is an ill-posed problem because theoretically there can be an infinite set of blur kernel and sharp image pairs that produce the blurry image. Therefore, knowing the blur kernel permits one to resolve the sharp image from the blurry image.
One solution is to use standard models for the PSF, such as a linear blur kernel for motion blurring, a circular PSF for defocus, and a Gaussian PSF for atmospheric blur. In addition to choosing a functional form, one must guess at the parameters or perform parametting. The edge profile method removes the guesswork and offers a way to quickly obtain a functional form, support size, and parameter values.
There are many PSF estimation methods in the literature. Fergus et al. describes one of the more accurate methods for estimating the blur kernel but it is also one of the most computationally demanding methods. (R. Fergus et al., “Removing camera shake from a single photograph”, SIGGRAPH, 2006, incorporated herein by reference). Their paper shows that a kernel can be estimated for blur due to camera shake by using natural image statistics together with a variational Bayes inference algorithm. Their algorithm has been demonstrated to work well in a number of cases. However, even for small images or when Fergus et al.'s method uses only a portion of an image, the solution is highly computationally intensive.
Cho et al. present a method for estimating the PSF using the same assumption of step edges before blurring. (Taeg S. Cho, Sylvain Paris, Berthold K. P. Horn, William T. Freeman, “Blur kernel estimation using the Radon transform”, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR'11, 2011, incorporated herein by reference). These authors use techniques of image reconstruction from projections for estimating the blur kernel. Their insight is that a line integral of the blur kernel can be considered as a convolution of the kernel with an image of an ideal line. They use the inverse Radon transform, which is used in image reconstruction, to reconstruct the PSF. However, in its current implementation the user must provide the edge profile length, which requires a parameter space search since this parameter value affects results. They also present a Maximum a Posteriori (“MAP”) method that incorporates the Radon transform as a prior to help solve for both the kernel and image. Their method is slow and is actually an iterative blind deconvolution technique, rather than a blur kernel estimation. Similarly, the edge profile method can be used as a prior within the MAP methodology.
Joshi, et al. detect edges in a blurry image and estimate the PSF under the assumption of a step edge before blurring; they use an iterative MAP approach for the blur kernel. (N. Joshi, R. Szeliski, and D. Kriegman, “PSF estimation using sharp edge prediction”, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR'08, 2008, incorporated herein by reference). Furthermore, they strive for a super-resolved blur kernel.
Much recent research in solving the image deblurring/restoration problem is in the area of blind deconvolution. Blind deconvolution attempts to iteratively solve for both the PSF and the sharp image from a blurry image by incorporating general knowledge of both the PSF and sharp image into an error function. These iterative methods are general, but complex and computationally expensive.
Non-blind deconvolution methods solve for the sharp image assuming an accurate blur kernel is known. Hence, methods of accurately estimating the blur kernel are required in order to provide input for the non-blind deconvolution methods.
Levin et al. describe a regularization technique using a sparse, natural image prior. (A. Levin et al., “Image and depth from a conventional camera with a coded aperture”, SIGGRAPH, 2007, incorporated herein by reference). This sparse method encourages the majority of image pixels to be piecewise smooth.